50 



Mr. I. Todhunter on Jacobi's Theorem. 



and therefore 



Hence 



rc?r Jo A 3 Jo A 5 



••••.<» 



The right-hand member is necessarily positive. If we put <r=r 2 , we may 

 write the equation thus, 



e?<r v 2 Jo A V A 2 / 



This shows that if p be kept constant, VP 3 continually increases with <r, 

 and therefore cannot vanish more than once, and therefore of course V 

 cannot vanish more than once. 



17. Ivory makes the following statement : — 



" Let V stand for the integral in the equation (7) ; and supposing that 

 p and r 2 vary so as always to satisfy that equation, we shall have 



dY' ' dY , n 

 —-dp + -—rdr — . 

 dp rdr 



Now, r 2 representing any positive quantity, we may conceive it to 

 increase from zero to be infinitely great ; in which case it follows from 



the nature of the function V, that during the whole increase —rdr 



rdr 



dY 



will be negative: wherefore the other term —dp will be positive; which 

 requires that^ decrease continually. 3 ' 



It is here in fact asserted that ^ is negative for such values of p and 



r as make V vanish ; this is, however, wrong, as equation (9) shows that 

 dY 



— is positive when V=0. The mistake was pointed out by Liouville in 



the memoir already cited, and the correction was accepted by Ivory in the 

 Philosophical Transactions for 1839, pages 265, 266. The mistake 

 vitiates the remainder of Ivory's memoir. 



The investigation of Art. 16 is taken in substance from Liouville's 

 memoir; he also demonstrates the proposition of Art. 14, but not in the 

 way which I have adopted. 



18. The extract given in the preceding article from Ivory's memoir 

 involves another error, which Liouville does not notice : the words "which 

 requires that p decrease continually," contain an arbitrary unproved asser- 

 tion. We have 



dp dr dr 



hence if ^ were negative, ~ Hr would be positive ; then ^ might be 

 dr a dp dr dr 



